Integrals based on monotone set functions

Klement, Erich Peter; Li, Jun; Mesiar, Radko; Pap, Endre

doi:10.1016/j.fss.2015.07.010

Cited by 61 publications

(14 citation statements)

References 59 publications

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“…In generalized measure and integral theory, there are several kinds of important nonlinear integrals, the Choquet integral [1], the Sugeno integral [2], the pan-integral [3] and the concave integral introduced by Lehrer [4], etc. (see also [5]). The pan-integral with respect to monotone measure μ relates to a commutative isotonic semiring ( ) , , R + ⊕ ⊗ , where ⊕ is a pan-addition and ⊗ is a pan-multiplication related by the distributivity property (see also [6]).…”

Section: Introductionmentioning

confidence: 85%

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Monotone Measures Defined by Pan-Integral

Wang¹,

Yu²,

Li³

2018

APM

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Given a pan-space () additivity, null-additivity, weak null-additivity and (S) property. Since the pan-integral based on a pair of pan-operations () , ⊕ ⊗ covers the Sugeno integral (based on () , ∨ ∧) and the Shilkret integral (based on () , ∨ ⋅), therefore, the previous related results for the Sugeno integral are covered by the results presented here, in the meantime, some special results related the Shilkret integral are also obtained.

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Section: Introductionmentioning

confidence: 85%

“…The following is three kinds of important nonlinear integrals [1] [2] [24], see also [5]. Consider a nonnegative real-valued measurable function…”

Section: Pan-operation and Integralsmentioning

confidence: 99%

Monotone Measures Defined by Pan-Integral

Wang¹,

Yu²,

Li³

2018

APM

Self Cite

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“…By Lemma 1 (property (G2)) it follows that the function t + f (t) is nondecreasing. Then, δ is nondecreasing by an easy observation based on Equation (14).…”

Section: Diagonals Of Symmetric Reflected Maxmin Copulasmentioning

confidence: 99%

“…It includes nonsymmetric copulas that are used for instance as more general fuzzy connectives [1,6]. Its associated measure may have a singular part, a fact of potential use in various copula-based integrals (see [14,15]). As we shall show in what follows the main idea for maxmin copulas is reflected to probabilistic extensions of semilinear copulas, so it might be worth while to study possible analogues to their extensions to various classes of constructions (cf.…”

Section: Introductionmentioning

confidence: 99%

Reflected maxmin copulas and modeling quadrant subindependence

Košir

Omladič

2020

Fuzzy Sets and Systems

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“…Integration with respect to general set functions has a long tradition, and current contributions on this subject are hot research topics in several branches of both pure and applied mathematics. In the Introduction of [22], an excellent historical review on this integration can be found; see also [24] for an overview of non-additive monotonic measures and their properties. There are a lot of mathematical developments related to non-additive integration; the references in both papers just mentioned provide a nice selections of works regarding this subject and its broad class of applications.…”

Section: Introductionmentioning

confidence: 99%

Choquet type L1-spaces of a vector capacity

Delgado

Pérez

2017

Fuzzy Sets and Systems

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Given a set function Λ with values in a Banach space X, we construct an integration theory for scalar functions with respect to Λ by using duality on X and Choquet scalar integrals. Our construction extends the classical Bartle-Dunford-Schwartz integration for vector measures. Since just the minimal necessary conditions on Λ are required, several L 1 -spaces of integrable functions associated to Λ appear in such a way that the integration map can be defined in them. We study the properties of these spaces and how they are related. We show that the behavior of the L 1 -spaces and the integration map can be improved in the case when X is an order continuous Banach lattice, providing new tools for (non-linear) operator theory and information science.

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Integrals based on monotone set functions

Cited by 61 publications

References 59 publications

Monotone Measures Defined by Pan-Integral

Monotone Measures Defined by Pan-Integral

Reflected maxmin copulas and modeling quadrant subindependence

Choquet type L1-spaces of a vector capacity

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